On the range of simple symmetric random walks on the line (Q1986024)

In this article, the authors study the random walk \((S_n(x))_{n \geq 0}\) defined by the dyadic expansion of the real number \(x \in [0,1]\). If \(x\) is chosen according to the Lebesgue measure on \([0,1]\), then \((S_n(x))_{n \geq 0}\) is a simple symmetric random walk. In that case, \textit{P. Révész} [Random walk in random and non-random environments. Singapore etc.: World Scientific (1990; Zbl 0733.60091)] showed that the range \(R_n(x) := \#\{S_j(x),0 \leq j \leq n \}\) of this random walk is almost surely of order \((n \log \log n)^{1/2}\) for \(n\) large enough. In this article, the authors compute the Hausdorff dimension of the set of points \(x \in [0,1]\) such that \(R_n(x) \sim c n^\gamma\) for \(c > 0\) and \(0 < \gamma \leq 1\). They proved the Hausdorff dimension to be \(1\) if \(\gamma < 1\), and that for all \(c \in (0,1)\): \[ \dim_H(\{x\in[0,1]:R_n(x)\sim cn\})=-\left(\tfrac{1+c}{2} \log_2\left( \tfrac{1+c}{2}\right) + \tfrac{1-c}{2} \log_2 \left(\tfrac{1-c}{2}\right) \right). \]